<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">prime</a>  iff <font color="Olive" title="b1">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">prime</a>  iff <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:65"/>
( <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">prime</a>  implies <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  implies <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">prime</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_2.html#D12" target="_self" title="FILTER_2:def.12">FILTER_2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">p9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D6" title="FILTER_0:def.6">FILTER_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> holds <br/> ( (  not <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  or <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) &amp; ( ( not <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) or <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">q9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  or <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  or <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) &amp; ( ( not <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  &amp; not <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) or <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) )</span><br/>



<a NAME="E2:65_1"/>
<font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c3">p9</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c4">q9</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:65_1"/>
( (  not <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  or <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  or <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) &amp; ( ( not <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  &amp; not <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) or <font color="Maroon" title="c3">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:65"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
for <font color="Olive" title="b1">p9</font>, <font color="Olive" title="b2">q9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> holds <br/> ( <font color="Olive" title="b1">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  iff ( <font color="Olive" title="b1">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  or <font color="Olive" title="b2">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_0.html#D6" title="FILTER_0:def.6">FILTER_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">prime</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="filter_2.html#D12" target="_self" title="FILTER_2:def.12">FILTER_2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> iff ( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> iff ( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> ) )</span><br/>



<a NAME="E3:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">q</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:65"/>
( <font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> iff ( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E23">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
